8 TOPOLOGICAL GROUPS 37 (i) Let H be a locally closed subgroup of G. Then H is a subgroup of G, and H is an open subgroup of H (12.2.3). Hence it is enough to prove the second assertion. Now if H has an interior point, then by translation it follows that every point of H is interior, and therefore H is open. Hence the left cosets xH are also open sets, and therefore (]H is open in G (because it is a union of cosets xH). Consequently H is closed in G. (ii) If G is Hausdorff and H is a discrete subgroup of G, then there exists a symmetric open neighborhood V of e such that V n H = {e}. If x e H, we have xV n H ^ 0. Now if y e xV n H, then x e yV and the set {y} is closed in the open set yV, because G is Hausdorff. Since yV n H = {y} (because y e H), we must have x = y, and therefore H = H. (12.8.8) If G is a connected group and V is a symmetric neighborhood of e, then G is equal to the union V00 of the sets V" for n > 0. The set V00 is clearly symmetric, and since VmV = Vm+n it follows that yooyoo <_ yoo Hence V°° is a subgroup of G. Since e is an interior point of V00, this subgroup is both open and closed (12.8.7) and therefore is the whole of G. (12.8.9) In a topological group G, the connected component K of the neutral element (3.19) is a closed normal subgroup (called the neutral component or identity component of G). For each x e G, the connected component ofx in G is xK = Kx. If a e K, the set a-1K is connected and contains e, hence K^K c K. This shows that K is a subgroup of G. It is invariant under all automorphisms of the topological group G, in particular under all inner automorphisms, hence K is normal in G. Also K is closed in G (3.19). Finally, the last assertion follows from the fact that left and right translations are homeomorphisms of G onto G. (12.8.10) If Gl9 G2 are two topological groups, then the product topol- ogy on the product group G = Gj x G2 is compatible with the group structure of G. For, identifying canonically the product spaces G x G and (G! x Gx) x (O2 x G2) (12.5), the mapping (C*i» x2)9 Oi, 72))>~>C*i.);i> *2J2) is continuous ((3.20.15) and (12.5)), and the mapping (xl9 x2)*-+(xil, x2l) is continuous for the same reason. The group Gt x G2, endowed with the product topology, is called the product of the topological groups G1 and G2. If G is a commutative topological group, the mapping (x, y)i-*xy is a continuous homomorphism of G x G into G.orff group, every discrete subgroup is closed.of the